mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.ZariskisMainTheorem

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
⊢ IsClosedImmersion f ↔ IsProper f ∧ Mono f

Tactic:
have (_ : Mono f) (_ : IsProper f) : LocallyQuasiFinite f := inferInstance

Goal state after:
X Y : Scheme
f : X ⟶ Y
this : Mono f → IsProper f → LocallyQuasiFinite f
⊢ IsClosedImmersion f ↔ IsProper f ∧ Mono f
Module: Mathlib.RingTheory.Ideal.CotangentBaseChange

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : CommRing T
inst✝ : Algebra R T
I : Ideal S
a : S →+* T ⊗[R] S := Algebra.TensorProduct.includeRight.toRingHom
⊢ Function.Surjective ⇑(tensorCotangentHom R T I)

Tactic:
intro x

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : CommRing T
inst✝ : Algebra R T
I : Ideal S
a : S →+* T ⊗[R] S := Algebra.TensorProduct.includeRight.toRingHom
x : (map Algebra.TensorProduct.includeRight.toRingHom I).Cotangent
⊢ ∃ a, (tensorCotangentHom R T I) a = x
Module: Mathlib.CategoryTheory.Bicategory.Adjunction.Adj

Goal state before:
pp: B : Type u
inst✝ : Bicategory B
a b c d : Adj B
α : a ⟶ b
⊢ (conjugateEquiv α.adj (α ≫ 𝟙 b).adj) (ρ_ α.l).hom = (λ_ α.r).symm.hom

Tactic:
simpa using conjugateEquiv_comp_id_right_apply α.adj α.adj (𝟙 _)

Goal state after:
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatRank

Goal state before:
pp: R S : CommRingCat
f : R ⟶ S
hf₁✝ : IsFinite (Spec.map f)
hf₂✝ : Flat (Spec.map f)
hf₁ : (CommRingCat.Hom.hom (appTop (Spec.map f))).Finite
hf₂ : (CommRingCat.Hom.hom (appTop (Spec.map f))).Flat
x✝ : ↥(Spec R)
⊢ (CommRingCat.Hom.hom (appTop (Spec.map f))).finrank ((Spec R).isoSpec.hom x✝) = (CommRingCat.Hom.hom f).finrank x✝

Tactic:
have : f = (Scheme.ΓSpecIso R).inv ≫ (Spec.map f).appTop ≫ (Scheme.ΓSpecIso S).hom := by simp

Goal state after:
R S : CommRingCat
f : R ⟶ S
hf₁✝ : IsFinite (Spec.map f)
hf₂✝ : Flat (Spec.map f)
hf₁ : (CommRingCat.Hom.hom (appTop (Spec.map f))).Finite
hf₂ : (CommRingCat.Hom.hom (appTop (Spec.map f))).Flat
x✝ : ↥(Spec R)
this : f = (ΓSpecIso R).inv ≫ appTop (Spec.map f) ≫ (ΓSpecIso S).hom
⊢ (CommRingCat.Hom.hom (appTop (Spec.map f))).finrank ((Spec R).isoSpec.hom x✝) = (CommRingCat.Hom.hom f).finrank x✝
Module: Mathlib.CategoryTheory.Bicategory.Adjunction.Mate

Goal state before:
pp: B : Type u
inst✝ : Bicategory B
a b c d e f : B
g₁ : a ⟶ c
g₂ : c ⟶ e
h₁ : b ⟶ d
h₂ : d ⟶ f
l₁ : a ⟶ b
r₁ : b ⟶ a
l₂ : c ⟶ d
r₂ : d ⟶ c
l₃ : e ⟶ f
r₃ : f ⟶ e
adj₁ : l₁ ⊣ r₁
adj₂ : l₂ ⊣ r₂
adj₃ : l₃ ⊣ r₃
α : g₁ ≫ l₂ ⟶ l₁ ≫ h₁
β : g₂ ≫ l₃ ⟶ l₂ ≫ h₂
⊢ 𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫
      r₁ ◁ g₁ ◁ g₂ ◁ adj₃.unit ⊗≫
        r₁ ◁ g₁ ◁ β ▷ r₃ ⊗≫
          (r₁ ≫ g₁) ◁ leftZigzag adj₂.unit adj₂.counit ▷ (h₂ ≫ r₃) ⊗≫
            r₁ ◁ α ▷ h₂ ▷ r₃ ⊗≫ adj₁.counit ▷ h₁ ▷ h₂ ▷ r₃ ⊗≫ 𝟙 ((h₁ ≫ h₂) ≫ r₃) =
    𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫
      r₁ ◁ (g₁ ≫ g₂) ◁ adj₃.unit ⊗≫
        r₁ ◁ ((α_ g₁ g₂ l₃).hom ≫ g₁ ◁ β ≫ (α_ g₁ l₂ h₂).inv ≫ α ▷ h₂ ≫ (α_ l₁ h₁ h₂).hom) ▷ r₃ ⊗≫
          adj₁.counit ▷ (h₁ ≫ h₂) ▷ r₃ ⊗≫ 𝟙 ((h₁ ≫ h₂) ≫ r₃)

Tactic:
adj₂.left_triangle

Goal state after:
B : Type u
inst✝ : Bicategory B
a b c d e f : B
g₁ : a ⟶ c
g₂ : c ⟶ e
h₁ : b ⟶ d
h₂ : d ⟶ f
l₁ : a ⟶ b
r₁ : b ⟶ a
l₂ : c ⟶ d
r₂ : d ⟶ c
l₃ : e ⟶ f
r₃ : f ⟶ e
adj₁ : l₁ ⊣ r₁
adj₂ : l₂ ⊣ r₂
adj₃ : l₃ ⊣ r₃
α : g₁ ≫ l₂ ⟶ l₁ ≫ h₁
β : g₂ ≫ l₃ ⟶ l₂ ≫ h₂
⊢ 𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫
      r₁ ◁ g₁ ◁ g₂ ◁ adj₃.unit ⊗≫
        r₁ ◁ g₁ ◁ β ▷ r₃ ⊗≫
          (r₁ ≫ g₁) ◁ ((λ_ l₂).hom ≫ (ρ_ l₂).inv) ▷ (h₂ ≫ r₃) ⊗≫
            r₁ ◁ α ▷ h₂ ▷ r₃ ⊗≫ adj₁.counit ▷ h₁ ▷ h₂ ▷ r₃ ⊗≫ 𝟙 ((h₁ ≫ h₂) ≫ r₃) =
    𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫
      r₁ ◁ (g₁ ≫ g₂) ◁ adj₃.unit ⊗≫
        r₁ ◁ ((α_ g₁ g₂ l₃).hom ≫ g₁ ◁ β ≫ (α_ g₁ l₂ h₂).inv ≫ α ▷ h₂ ≫ (α_ l₁ h₁ h₂).hom) ▷ r₃ ⊗≫
          adj₁.counit ▷ (h₁ ≫ h₂) ▷ r₃ ⊗≫ 𝟙 ((h₁ ≫ h₂) ≫ r₃)
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatRank

Goal state before:
pp: R S : CommRingCat
φ : R ⟶ S
inst✝¹ : Flat (Spec.map φ)
inst✝ : IsFinite (Spec.map φ)
h : 1 ≤ finrank (Spec.map φ)
⊢ Function.Surjective ⇑(Spec.map φ)

Tactic:
intro x

Goal state after:
R S : CommRingCat
φ : R ⟶ S
inst✝¹ : Flat (Spec.map φ)
inst✝ : IsFinite (Spec.map φ)
h : 1 ≤ finrank (Spec.map φ)
x : ↥(Spec R)
⊢ ∃ a, (Spec.map φ) a = x
Module: Mathlib.RingTheory.KrullDimension.NonZeroDivisors

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
r : R
hr : r ∈ R⁰
hr' : ¬Ideal.span {r} = ⊤
this : Nonempty ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))
⊢ ringKrullDim (R ⧸ Ideal.span {r}) + 1 ≤ ringKrullDim R

Tactic:
have := Ideal.Quotient.nontrivial_iff.mpr hr'

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
r : R
hr : r ∈ R⁰
hr' : ¬Ideal.span {r} = ⊤
this✝ : Nonempty ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))
this : Nontrivial (R ⧸ Ideal.span {r})
⊢ ringKrullDim (R ⧸ Ideal.span {r}) + 1 ≤ ringKrullDim R
Module: Mathlib.RingTheory.OrderOfVanishing.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
a : R
I : Ideal R
⊢ Function.Surjective ⇑(Submodule.factor ⋯)

Tactic:
exact Submodule.factor_surjective <|
    Submodule.singleton_set_smul I a ▸ Submodule.smul_le_span {a} I

Goal state after:
Module: Mathlib.RingTheory.OrderOfVanishing.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
a x : R
this : span {a * x} ≤ span {x}
⊢ Module.length R (R ⧸ span {x}) ≤ Module.length R (R ⧸ span {a * x})

Tactic:
let g : (R ⧸ Ideal.span {a * x}) →ₗ[R] (R ⧸ Ideal.span {x}) := Submodule.factor this

Goal state after:
R : Type u_1
inst✝ : CommRing R
a x : R
this : span {a * x} ≤ span {x}
g : R ⧸ span {a * x} →ₗ[R] R ⧸ span {x} := Submodule.factor this
⊢ Module.length R (R ⧸ span {x}) ≤ Module.length R (R ⧸ span {a * x})
Module: Mathlib.RingTheory.OrderOfVanishing.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : CommRing R
inst✝¹ : Module R M
inst✝ : Nontrivial R
⊢ ∀ (x y : R),
    (if x * y ∈ nonZeroDivisors R then ENat.recTopCoe 0 (fun x ↦ ↑(Multiplicative.ofAdd ↑x)) (ord R (x * y)) else 0) =
      (if x ∈ nonZeroDivisors R then ENat.recTopCoe 0 (fun x ↦ ↑(Multiplicative.ofAdd ↑x)) (ord R x) else 0) *
        if y ∈ nonZeroDivisors R then ENat.recTopCoe 0 (fun x ↦ ↑(Multiplicative.ofAdd ↑x)) (ord R y) else 0

Tactic:
intro x

Goal state after:
R : Type u_1
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : CommRing R
inst✝¹ : Module R M
inst✝ : Nontrivial R
x : R
⊢ ∀ (y : R),
    (if x * y ∈ nonZeroDivisors R then ENat.recTopCoe 0 (fun x ↦ ↑(Multiplicative.ofAdd ↑x)) (ord R (x * y)) else 0) =
      (if x ∈ nonZeroDivisors R then ENat.recTopCoe 0 (fun x ↦ ↑(Multiplicative.ofAdd ↑x)) (ord R x) else 0) *
        if y ∈ nonZeroDivisors R then ENat.recTopCoe 0 (fun x ↦ ↑(Multiplicative.ofAdd ↑x)) (ord R y) else 0
Module: Mathlib.RingTheory.GradedAlgebra.FiniteType

Goal state before:
pp: S : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : CommRing S
inst✝³ : SetLike σ S
inst✝² : AddSubgroupClass σ S
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
inst✝ : Algebra.FiniteType (↥(𝒜 0)) S
F : Finset S
hF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤
ι₀ : Type (max u_1 u_3) := (x : ↥F) × ↥(DFinsupp.support ((DirectSum.decompose 𝒜) ↑x))
x : ι₀ → S := fun i ↦ ↑(((DirectSum.decompose 𝒜) ↑i.fst) ↑i.snd)
⊢ ∃ s, Algebra.adjoin ↥(𝒜 0) ↑s = ⊤ ∧ ∀ i ∈ s, SetLike.IsHomogeneousElem 𝒜 i

Tactic:
refine ⟨Finset.univ.image x, ?_, by simpa using fun f ↦ ⟨_, (DirectSum.decompose 𝒜 f.1 f.2).2⟩⟩

Goal state after:
S : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : CommRing S
inst✝³ : SetLike σ S
inst✝² : AddSubgroupClass σ S
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
inst✝ : Algebra.FiniteType (↥(𝒜 0)) S
F : Finset S
hF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤
ι₀ : Type (max u_1 u_3) := (x : ↥F) × ↥(DFinsupp.support ((DirectSum.decompose 𝒜) ↑x))
x : ι₀ → S := fun i ↦ ↑(((DirectSum.decompose 𝒜) ↑i.fst) ↑i.snd)
⊢ Algebra.adjoin ↥(𝒜 0) ↑(Finset.image x Finset.univ) = ⊤
Module: Mathlib.RingTheory.GradedAlgebra.FiniteType

Goal state before:
pp: S : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : CommRing S
inst✝³ : SetLike σ S
inst✝² : AddSubgroupClass σ S
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
inst✝ : Algebra.FiniteType (↥(𝒜 0)) S
F : Finset S
hF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤
ι₀ : Type (max u_1 u_3) := (x : ↥F) × ↥(DFinsupp.support ((DirectSum.decompose 𝒜) ↑x))
x : ι₀ → S := fun i ↦ ↑(((DirectSum.decompose 𝒜) ↑i.fst) ↑i.snd)
s : S
hs : s ∈ ↑F
⊢ s ∈ ↑(Algebra.adjoin ↥(𝒜 0) ↑(Finset.image x Finset.univ))

Tactic:
← DirectSum.sum_support_decompose 𝒜 s

Goal state after:
S : Type u_1
σ : Type u_2
ι : Type u_3
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : CommRing S
inst✝³ : SetLike σ S
inst✝² : AddSubgroupClass σ S
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
inst✝ : Algebra.FiniteType (↥(𝒜 0)) S
F : Finset S
hF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤
ι₀ : Type (max u_1 u_3) := (x : ↥F) × ↥(DFinsupp.support ((DirectSum.decompose 𝒜) ↑x))
x : ι₀ → S := fun i ↦ ↑(((DirectSum.decompose 𝒜) ↑i.fst) ↑i.snd)
s : S
hs : s ∈ ↑F
⊢ ∑ i ∈ DFinsupp.support ((DirectSum.decompose 𝒜) s), ↑(((DirectSum.decompose 𝒜) s) i) ∈
    ↑(Algebra.adjoin ↥(𝒜 0) ↑(Finset.image x Finset.univ))
Module: Mathlib.RingTheory.GradedAlgebra.Radical

Goal state before:
pp: ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : LinearOrder ι
inst✝³ : IsOrderedCancelAddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
⊢ ∀ {x y : A}, x * y ∈ I → x ∈ I ∨ y ∈ I

Tactic:
intro x

Goal state after:
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : LinearOrder ι
inst✝³ : IsOrderedCancelAddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x : A
⊢ ∀ {y : A}, x * y ∈ I → x ∈ I ∨ y ∈ I
Module: Mathlib.RingTheory.GradedAlgebra.Radical

Goal state before:
pp: ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : LinearOrder ι
inst✝³ : IsOrderedCancelAddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x y : A
hxy : x * y ∈ I
rid₁ : x ∉ I
rid₂ : y ∉ I
set₁ : Finset ι := {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}
set₁_eq : set₁ = {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}
set₂ : Finset ι := {i ∈ DFinsupp.support ((decompose 𝒜) y) | (proj 𝒜 i) y ∉ I}
set₂_eq : set₂ = {i ∈ DFinsupp.support ((decompose 𝒜) y) | (proj 𝒜 i) y ∉ I}
⊢ ∀ x ∉ I, {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}.Nonempty

Tactic:
intro x

Goal state after:
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : LinearOrder ι
inst✝³ : IsOrderedCancelAddMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x✝ y : A
hxy : x✝ * y ∈ I
rid₁ : x✝ ∉ I
rid₂ : y ∉ I
set₁ : Finset ι := {i ∈ DFinsupp.support ((decompose 𝒜) x✝) | (proj 𝒜 i) x✝ ∉ I}
set₁_eq : set₁ = {i ∈ DFinsupp.support ((decompose 𝒜) x✝) | (proj 𝒜 i) x✝ ∉ I}
set₂ : Finset ι := {i ∈ DFinsupp.support ((decompose 𝒜) y) | (proj 𝒜 i) y ∉ I}
set₂_eq : set₂ = {i ∈ DFinsupp.support ((decompose 𝒜) y) | (proj 𝒜 i) y ∉ I}
x : A
⊢ x ∉ I → {i ∈ DFinsupp.support ((decompose 𝒜) x) | (proj 𝒜 i) x ∉ I}.Nonempty
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
ι : Type u_2
f : ι → A
hfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n
hf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤
⊢ (HomogeneousIdeal.irrelevant 𝒜).toIdeal ≤ Ideal.span (Set.range f)

Tactic:
intro x

Goal state after:
σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
ι : Type u_2
f : ι → A
hfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n
hf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤
x : A
⊢ x ∈ (HomogeneousIdeal.irrelevant 𝒜).toIdeal → x ∈ Ideal.span (Set.range f)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: case a
σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f✝ : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
⊢ (HomogeneousIdeal.irrelevant 𝒜).toIdeal ≤ Ideal.span (Set.range fun x ↦ x.snd.fst)

Tactic:
intro x

Goal state after:
case a
σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f✝ : A →+* ↑Γ(X, ⊤)
x✝ x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
x : A
⊢ x ∈ (HomogeneousIdeal.irrelevant 𝒜).toIdeal → x ∈ Ideal.span (Set.range fun x ↦ x.snd.fst)
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M✝ : ModuleCat ↑R
M N : (Spec R).Modules
f : modulesSpecToSheaf.obj M ⟶ modulesSpecToSheaf.obj N
U : (Opens ↥(Spec R))ᵒᵖ
t : ↑((Spec R).ringCatSheaf.obj.obj U)
m : ↑(M.val.1 U)
⊢ ∀ x ∈ unop U,
    ∃ V,
      ∃ (hV : V ≤ unop U),
        x ∈ V ∧
          (ConcreteCategory.hom ((modulesSpecToSheaf.obj N).obj.map (homOfLE hV).op))
              ((ModuleCat.Hom.hom (f.hom.app U)).toFun (t • m)) =
            (ConcreteCategory.hom ((modulesSpecToSheaf.obj N).obj.map (homOfLE hV).op))
              ((RingHom.id ↑((Spec R).ringCatSheaf.obj.obj U)) t • (ModuleCat.Hom.hom (f.hom.app U)).toFun m)

Tactic:
intro x

Goal state after:
R : CommRingCat
M✝ : ModuleCat ↑R
M N : (Spec R).Modules
f : modulesSpecToSheaf.obj M ⟶ modulesSpecToSheaf.obj N
U : (Opens ↥(Spec R))ᵒᵖ
t : ↑((Spec R).ringCatSheaf.obj.obj U)
m : ↑(M.val.1 U)
x : ↥(Spec R)
⊢ x ∈ unop U →
    ∃ V,
      ∃ (hV : V ≤ unop U),
        x ∈ V ∧
          (ConcreteCategory.hom ((modulesSpecToSheaf.obj N).obj.map (homOfLE hV).op))
              ((ModuleCat.Hom.hom (f.hom.app U)).toFun (t • m)) =
            (ConcreteCategory.hom ((modulesSpecToSheaf.obj N).obj.map (homOfLE hV).op))
              ((RingHom.id ↑((Spec R).ringCatSheaf.obj.obj U)) t • (ModuleCat.Hom.hom (f.hom.app U)).toFun m)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: case a
σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
r : A
n : ℕ
hn : 0 < n
hr : r ∈ 𝒜 n
⊢ fromOfGlobalSections 𝒜 f hf ⁻¹ᵁ basicOpen 𝒜 r ≤ X.basicOpen (f r)

Tactic:
intro x

Goal state after:
case a
σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
r : A
n : ℕ
hn : 0 < n
hr : r ∈ 𝒜 n
x : ↥X
⊢ x ∈ fromOfGlobalSections 𝒜 f hf ⁻¹ᵁ basicOpen 𝒜 r → x ∈ X.basicOpen (f r)
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
U : (Spec R).Opens
f : ↑R
hle : U ≤ PrimeSpectrum.basicOpen f
⊢ IsSMulRegular (↑Γ(M, U)) f

Tactic:
intro x

Goal state after:
R : CommRingCat
M : (Spec R).Modules
U : (Spec R).Opens
f : ↑R
hle : U ≤ PrimeSpectrum.basicOpen f
x : ↑Γ(M, U)
⊢ ∀ ⦃a₂ : ↑Γ(M, U)⦄, (fun x ↦ f • x) x = (fun x ↦ f • x) a₂ → x = a₂
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: case a
σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
r : A
n : ℕ
hn : 0 < n
hr : r ∈ 𝒜 n
⊢ X.basicOpen (f r) ≤ fromOfGlobalSections 𝒜 f hf ⁻¹ᵁ basicOpen 𝒜 r

Tactic:
intro x

Goal state after:
case a
σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
r : A
n : ℕ
hn : 0 < n
hr : r ∈ 𝒜 n
x : ↥X
⊢ x ∈ X.basicOpen (f r) → x ∈ fromOfGlobalSections 𝒜 f hf ⁻¹ᵁ basicOpen 𝒜 r
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ℕ
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ℕ
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
hxd' : ∀ (i : ι'), 0 < dv i
a : ℕ
ai : ι' → ℕ
hai : ∑ i, ai i • dv i = a • d
| ∑ i, ai i • dv i

Tactic:
2,

Goal state after:
A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ℕ
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ℕ
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
hxd' : ∀ (i : ι'), 0 < dv i
a : ℕ
ai : ι' → ℕ
hai : ∑ i, ai i • dv i = a • d
| fun i ↦ ai i • dv i
Module: Mathlib.RingTheory.LocalRing.LocalSubring

Goal state before:
pp: K : Type u_3
inst✝¹ : Field K
A : Subring K
P : Ideal ↥A
inst✝ : P.IsPrime
⊢ A ≤ (ofPrime A P).toSubring

Tactic:
intro x

Goal state after:
K : Type u_3
inst✝¹ : Field K
A : Subring K
P : Ideal ↥A
inst✝ : P.IsPrime
x : K
⊢ x ∈ A → x ∈ (ofPrime A P).toSubring
Module: Mathlib.RingTheory.LocalRing.LocalSubring

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
K : Type u_3
inst✝¹ : Field K
A : Subring K
P : Ideal ↥A
inst✝ : P.IsPrime
⊢ Function.Injective ⇑(IsLocalization.liftAlgHom ⋯)

Tactic:
intro x

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
K : Type u_3
inst✝¹ : Field K
A : Subring K
P : Ideal ↥A
inst✝ : P.IsPrime
x : Localization.AtPrime P
⊢ ∀ ⦃a₂ : Localization.AtPrime P⦄, (IsLocalization.liftAlgHom ⋯) x = (IsLocalization.liftAlgHom ⋯) a₂ → x = a₂
Module: Mathlib.RingTheory.Valuation.LocalSubring

Goal state before:
pp: case pq
K : Type u_3
inst✝ : Field K
s : Set K
V : ValuationSubring K
this : IsIntegrallyClosedIn (↥V.toSubring) K
⊢ s ⊆ ↑V.toSubring ↔ (integralClosure (↥(Subring.closure s)) K).toSubring ≤ V.toSubring

Tactic:
rw [Subring.integralClosure_subring_le_iff]

Goal state after:
case pq
K : Type u_3
inst✝ : Field K
s : Set K
V : ValuationSubring K
this : IsIntegrallyClosedIn (↥V.toSubring) K
⊢ s ⊆ ↑V.toSubring ↔ Subring.ofClass (Subring.closure s) ≤ V.toSubring
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: case surj.refine_2
A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x).asHomogeneousIdeal
⊢ (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (f ^ i * (↑a * b ^ m)) =
    (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (f ^ i * b * (↑a * b ^ (m - 1)))

Tactic:
mul_mul_mul_comm,

Goal state after:
case surj.refine_2
A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x).asHomogeneousIdeal
⊢ (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (f ^ i * (↑a * b ^ m)) =
    (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (f ^ i * ↑a * (b * b ^ (m - 1)))
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
y z : NumDenSameDeg 𝒜 (Submonoid.powers f)
e :
  (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) (HomogeneousLocalization.mk y) =
    (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) (HomogeneousLocalization.mk z)
⊢ ∃ i, ∃ c ∈ 𝒜 i, c ∉ (↑x).asHomogeneousIdeal ∧ c * (↑z.den * ↑y.num) = c * (↑y.den * ↑z.num)

Tactic:
apply_fun HomogeneousLocalization.val at e

Goal state after:
A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
y z : NumDenSameDeg 𝒜 (Submonoid.powers f)
e :
  HomogeneousLocalization.val
      ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) (HomogeneousLocalization.mk y)) =
    HomogeneousLocalization.val
      ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) (HomogeneousLocalization.mk z))
⊢ ∃ i, ∃ c ∈ 𝒜 i, c ∉ (↑x).asHomogeneousIdeal ∧ c * (↑z.den * ↑y.num) = c * (↑y.den * ↑z.num)
Module: Mathlib.AlgebraicGeometry.ValuativeCriterion

Goal state before:
pp: X✝ Y : Scheme
f : X✝ ⟶ Y
Y' X : Scheme
⊢ ∀ {Y'_1 S : Scheme} {f : Y' ⟶ S} {g : X ⟶ S} {f' : Y'_1 ⟶ X} {g' : Y'_1 ⟶ Y'},
    IsPullback f' g' g f → Existence g → Existence g'

Tactic:
X'

Goal state after:
X✝ Y : Scheme
f : X✝ ⟶ Y
Y' X X' : Scheme
⊢ ∀ {S : Scheme} {f : Y' ⟶ S} {g : X ⟶ S} {f' : X' ⟶ X} {g' : X' ⟶ Y'},
    IsPullback f' g' g f → Existence g → Existence g'
Module: Mathlib.AlgebraicGeometry.Sites.Small

Goal state before:
pp: P : MorphismProperty Scheme
S : Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
inst✝ : P.RespectsIso
X : Over S
R : Sieve X
⊢ R ∈ (overGrothendieckTopology P S) X ↔ R ∈ (overPretopology P S).toGrothendieck X

Tactic:
rw [GrothendieckTopology.mem_over_iff]

Goal state after:
P : MorphismProperty Scheme
S : Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
inst✝ : P.RespectsIso
X : Over S
R : Sieve X
⊢ (Sieve.overEquiv X) R ∈ (grothendieckTopology P) X.left ↔ R ∈ (overPretopology P S).toGrothendieck X